In the previous subpart, you gathered evidence of some properties of angles. But evidence alone doesn't explain why something is true, or even mean that it is true. Mathematics requires a reasoned argument that is general, not about a specific set of lines.

When two lines intersect, the vertical angles (angles opposite each other) have the same measure.

Problem B5

In this problem, you will look at an explanation for why vertical angles have the same measure.

a.

m1 + m2 = 180°. Why?

b.

Also m3 + m2 = 180°. Why?

c.

So m1 must equal m3. Why?

d.

What other pair of angles is equal in measure? Why?

When two parallel lines both intersect a third line, corresponding angles (angles in the same relative positions, like angles 1 and 7 or angles 3 and 5 in the picture below) have the same measure.

One way to understand this is to imagine sliding a copy of the picture above along line j until line k sits on top of line l. Note 3

Now 1 sits exactly where 7 used to be, 3 sits exactly where 5 used to be, and so on.

To prove that corresponding angles are congruent, we could add another line segment, , parallel to line j. By doing so we have created a parallelogram, and thus we know that the adjacent angles of a parallelogram (in this case 2 and 7) equal 180°.

So, to prove that 1 and 7 are congruent, we write the following:

1 + 2 = 180° (because they form a straight line)2 + 7 = 180° (because they are adjacent angles of a parallelogram)

It follows that 1 + 2 = 2 + 7 = 180°.

And thus, 1 = 7

Alternate interior angles (angles on opposite sides of the transversal, and between the parallel lines, like 7 and 3 or 2 and 8), also have the same measure.

Problem B6

Create an argument to explain why the above statement about alternate interior angles would be true. You may want to use the facts about corresponding angles and vertical angles, or you may come up with another explanation.